This document provides a preliminary report of the study Gateways To Advance Mathematical Thinking (GAMT) run by Educational Development Center, Inc. (EDC). The study was designed to see what types of reasoning students who have recently completed a linear algebra course apply to problems in algebraic thinking. Student interviews were used as the method of research with six questions on dot products, systems of linear equations, and linear transformations. In analyzing the data, six main issues were chosen as the focus of inquiry: dot product; relationship between a linear transformation and its matrix; solving a system of linear equations; meaning, number, and nature of the solution to a system of linear equations; visualization; and linear thinking: properties of a linear transformation. Preliminary results are based on early work with "dot product" and "meaning, number, and nature of the solution to a system of linear equations." Most of the paper consists of transcripts of interviews with students during their attempts to solve problems. Annotation of the interviews and analysis is also included. (MM)